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###### The Evans–Young method

Local topographic variables are functions of partial derivatives of elevation , , , , . There are methods for computing r, t, s, p, and q from digital elevation models (DEMs) based on plane square grids. These methods use an approximation of partial derivatives by finite differences using the 3x3 plane square-gridded moving window.

In the EvansYoung method, the second-order polynomial

is fitted by the least-squares approach to the nine points of the 3x3 square-spaced window with a grid spacing of w:

The Cartesian coordinates and elevations of the topographic surface are known for the window points (-w, w, z1), (0, w, z2), (w, w, z3), (-w, 0, z4), (0, 0, z5), (w, 0, z6), (-w, -w, z7), (0, -w, z8) and (w, -w, z9). For the point (0, 0, z5), the polynomial coefficients (which are partial derivatives of elevation) are estimated by the following formulae:

,

,

,

,

.

Moving the 3x3 window along a DEM, one can calculate values of r, t, s, p, and q (and so values of local morphometric variables) for all points of the plane square-gridded DEM, except for boundary rows and columns.

References

Evans, I.S., 1979. Statistical Characterization of Altitude Matrices by Computer. An Integrated System of Terrain Analysis and Slope Mapping. The Final Report on Grant DA-ERO-591-73-G0040. Department of Geography, University of Durham, Durham, 192 p.

Young, M., 1978. Statistical Characterization of Altitude Matrices by Computer. Terrain Analysis: Program Documentation. Report 5 on Grant DA-ERO-591-73-G0040. Department of Geography, University of Durham, Durham, 18 p.

For details and examples, see:

 DIGITAL TERRAIN ANALYSIS IN SOIL SCIENCE AND GEOLOGY   2nd revised edition     I.V. Florinsky   Elsevier / Academic Press, 2016 Amsterdam, 486 p.   ISBN 978-0-12-804632-6